\({27^{5-{x^2}}}-{3^{{x^2}-1}} = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{3^{15-3{x^2}}} = {3^{{x^2}-1}}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,15-3{x^2} = {x^2}-1\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,4{x^2} = 16\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{x^2}-4 = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left( {x-2} \right)\left( {x + 2} \right) = 0\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x-2 = 0,}\\{x + 2 = 0\,}\end{array}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = 2,\,\,\,}\\{x = -2.}\end{array}} \right.} \right.\)
Произведение корней: \(-2 \cdot 2 = -4\).
Ответ: \(-4.\)